The Direct Discontinuous Galerkin Method And Efficient Implicit Time-marching Scheme For Sobolev Equation Based On Space-time Element

In this paper,the direct discontinuous Galerkin method is used to discretize the Sobolev Equation in one-dimensional space,and the numerical stability of the scheme is proved.In order to improve the calculation efficiency of implicit time marching,the predictor-corrector approach based on Lax-Wendroff scheme is adopted in this paper.The corrector step is an improved forward Euler step.The locally-implicit method and regionally-implicit method are used to calculate the predictor step respectively,and thus the space-time discrete scheme in one-dimensional case is obtained.The discontinuous finite element space is composed of piecewise polynomials of degree 6)? 1.Numerical experiments show the effectiveness of the algorithm: all numerical schemes can achieve high-order accuracy,and compared with the explicit time-marching scheme,the implicit-time marching scheme can achieve higher computational efficiency.